Only the word Mawr, derived from M-W-R, is mentioned in the verse about the Day of Judgment, not Maira itself. The roots of many good names are used in such verses as well, so this is not a reason to reject this name.
There are two words that are pronounced as Maira (the same way in British English "lore" and "law" sound like the same word). One of the words basically means "fast-moving", "light", and the other means "bringer of food". These are two different words, when you choose one of them (say, "fast-moving"), then the other meaning(bringer of food) is excluded.
Islamically these two words are acceptable names. But whether you like the meaning or not is up to you. There are many girl names with better meanings, and many girl names with similar general meanings (Aisha means "one who lives", which is not a special/spiritual meaning).
So you can keep this name if you like it, there is nothing wrong with it.
Tania doesn't sound like any Arabic/Quranic word I know of. It can be another spelling of Tanya, a Russian name the is derived from Tatiana, an ancient Roman name. It is also a Sanskrit word meaning "daughter". If said with a short "a" at the beginning and a long "a" at the end, as in (TU)ck + (N)ame + (YA)rn, it becomes a popular Kurdish girl name meaning "alone", "unique."
Therefore Tania is not an Islamic name but since it doesn't have a bad meaning in Sanskrit or Kurdish, Muslims can use it.